Bell Work

                                                             If you want to have the best chances of getting a red gumball from a gumball machine, is it better if the machine is full of gumballs or half empty?  How do the chances of getting an ace in a deck of playing cards change if you have three or four decks of cards to choose from instead of only one deck?  In this lesson, you will think about the size of the sample space (the collection of all possible outcomes of an event).  Think about these questions as you work today: How has the “whole” or total changed? How has the “portion” or part we are interested in changed? Has the event become more or less likely?

1-75. Your team will be given a bag containing a set of colored blocks or counters. Alternatively, use the 1-75 Student eTool which contains a bag with 1 yellow, 2 red, 4 green, and 5 blue blocks. Each team will receive a bag that is identical to yours. a. Look at the blocks in your bag.  If you were to reach into the bag and select one block without looking, what is the likelihood that it would be: i. Red? ii. Green? iii. Blue? iv. Orange? b. Do your answers for part (a) represent theoretical or experimental probabilities?  Justify your response.

1-76.  If you were to select one block from the bag 12 times, replacing the block you drew between each selection, how many of those times would you expect to have selected a blue block?  What if you drew 24 times?  Discuss both situations with your team and explain your answers.  

1-77. DOUBLING BAGS                     Now imagine that you and another team have combined the blocks from both of your bags into one bag. a. Do you think the larger sample space will change the likelihood of drawing blocks of different colors?  Discuss this with your team and be ready to explain your ideas to the class.  b. How many total blocks are there in the bag now?  How many are there of each color?  

1-77 cont. c. Work with your team to find the theoretical probability for selecting each color of block in the combined bags.   d. Has the probability for drawing each different-colored block changed?  e. If you were to make 12 draws from the combined bag, replacing the block between draws, how many times would you expect to draw a blue block?  Explain why your answer makes sense.

1-78. In problems 1-75 through 1-77, even though you combined bags or changed the number of selections you made, the probability of drawing a blue block remained the same. Do you think the probabilities would change if you combined three bags?  Why or why not? What change do you think you could make in order to increase the chances of choosing a blue block?  Explain your reasoning.

Probability Vocabulary and Definitions Methods and Meanings: Probability Vocabulary and Definitions Outcome: Any possible or actual result of the action considered, such as rolling a 5 on a standard number cube or getting tails when flipping a coin. Event:  A desired (or successful) outcome or group of outcomes from an experiment, such as rolling an even number on a standard number cube. Sample Space:  All possible outcomes of a situation.  For example, the sample space for flipping a coin is heads and tails; rolling a standard number cube has six possible outcomes (1, 2, 3, 4, 5, and 6). Probability: The likelihood that an event will occur.  Probabilities may be written as fractions, decimals, or percents.  An event that is guaranteed to happen has a probability of 1, or 100%.  An event that has no chance of happening has a probability of 0, or 0%.  Events that “might happen” have probabilities between 0 and 1 or between 0% and 100%.  In general, the more likely an event is to happen, the greater its probability. 

Methods and Meanings, cont. Experimental Probability:  The probability based on data collected in experiments. Experimental Probability =  Theoretical Probability is a calculated probability based on the possible outcomes when they all have the same chance of occurring. Theoretical Probability = In the context of probability, “successful” usually means a desired or specified outcome (event), such as rolling a 2 on a number cube (probability of 1 6 ).  To calculate the probability of rolling a 2, first figure out how many possible outcomes there are.  Since there are six faces on the number cube, the number of possible outcomes is 6.  Of the six faces, only one of the faces has a 2 on it.  Thus, to find the probability of rolling a 2, you would write: or  or approximately 16.7%.

Practice 1. A spinner is divided into four equal sections numbered 2, 4, 6, and 8. What is the probability of spinning an 8? 2. There are 15 marbles in a bag; 5 blue, 6 yellow, and 4 green. What is the probability of getting a blue marble? 3. Joe flipped a coin 50 times. What is the probability of Joe landing on heads? 4. There are 12 marbles in a bag: 2 clear, 4 green, 5 yellow, and 1 blue. If one marble is chosen randomly from the bag, what is the probability that it will be yellow? 5. If you roll a fair, 6-sided number cube, what is P(3) , that is, the probability that you will roll a 3?